Bulletin of the
Korean Mathematical Society BKMS

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. In this paper we show that a finitely generated $R$-module $M$ of dimension $d$ is Cohen-Macaulay if and only if there exists a proper ideal $I$ of $R$ such that ${\rm depth}(M/I^nM)=d$ for $n\gg0$. Also we show that, if $\dim(R)=d$ and $I_1\subset\cdots\subset I_n$ is a chain of ideals of $R$ such that $R/I_k$ is maximal Cohen-Macaulay for all $k$, then $n\leq \ell_R(R/(a_1,\ldots,a_d)R)$ for every system of parameters $a_1,\ldots,a_d$ of $R$. Also, in the case where $\dim(R)=2$, we prove that the ideal transform $\rm D_{\mathfrak m}(R/\mathfrak p)$ is minimax balanced big Cohen-Macaulay, for every $\mathfrak p \in {\rm Assh}_R(R)$, and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

Keywords: balanced big Cohen-Macaulay modules, Cohen-Macaulay modules, local cohomology modules, quasi regular sequences

MSC numbers: 13D45, 14B15, 13E05